20 Theta Functions Properties 20.9 Relations to Other Functions 20.11 Generalizations and Analogs

§20.10 Integrals

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Keywords:: integrals, theta functions
Permalink:: http://dlmf.nist.gov/20.10
See also:: Annotations for Ch.20

§20.10(i) Mellin Transforms with respect to the Lattice Parameter
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
§20.10(iii) Compendia

§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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Keywords:: Mellin transform with respect to lattice parameter, theta functions
Notes:: For (20.10.1) and (20.10.3) use §20.7(viii) with appropriate changes of integration variable. For (20.10.2) use (20.2.3) with $z=0$ , $\tau=\mathrm{i}t$ , Bellman (1961, pp. 28–32), Koblitz (1993, pp. 70–75), and/or Titchmarsh (1986b, §2.6).
Referenced by:: §20.9(iii)
Permalink:: http://dlmf.nist.gov/20.10.i
Change (effective with 1.1.5):: The general constraint $\Re s>2$ at the start of this subsection has been made more specific in (20.10.1), (20.10.2) and (20.10.3).
See also:: Annotations for §20.10 and Ch.20

20.10.1		$\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle\|ix^{2}\right)\,\mathrm{d}x=(2% ^{s}-1)\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
$\Re s>1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part Keywords: Mellin transform Referenced by: §20.10(i), Erratum (V1.1.5) for §20.10(i), Erratum (V1.2.8) for Equation (20.10.1) Permalink: http://dlmf.nist.gov/20.10.E1 Encodings: TeX, pMML, png Simplification (effective with 1.2.8): Originally the right-hand side contained a prefactor given by $2^{s}(1-2^{-s})$ . This was replaced by $(2^{s}-1)$ . Modification (effective with 1.1.5): Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{2}\left(0\middle\|ix^{2}\right)=x^{-1}\theta_{4}\left(0\middle\|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.31) and (20.4.5). See also: Annotations for §20.10(i), §20.10 and Ch.20

20.10.2		$\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle\|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
$\Re s>1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part Keywords: Mellin transform Referenced by: §20.10(i), Erratum (V1.1.5) for §20.10(i) Permalink: http://dlmf.nist.gov/20.10.E2 Encodings: TeX, pMML, png Modification (effective with 1.1.5): Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{3}\left(0\middle\|ix^{2}\right)=x^{-1}\theta_{3}\left(0\middle\|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.32) and (20.4.4). See also: Annotations for §20.10(i), §20.10 and Ch.20

20.10.3		$\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle\|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
$\Re s>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part Keywords: Mellin transform Referenced by: §20.10(i), Erratum (V1.1.5) for §20.10(i) Permalink: http://dlmf.nist.gov/20.10.E3 Encodings: TeX, pMML, png Modification (effective with 1.1.5): Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>0$ because $1-\theta_{4}\left(0\middle\|ix^{2}\right)=1-x^{-1}\theta_{2}\left(0\middle\|ix^{% -2}\right)\sim 1$ as $x\to 0+$ , which follows from (20.7.33) and (20.4.3). See also: Annotations for §20.10(i), §20.10 and Ch.20

Here $\zeta\left(s\right)$ again denotes the Riemann zeta function (§25.2).

For further results see Oberhettinger (1974, pp. 157–159).

§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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Keywords:: Laplace transform with respect to lattice parameter, theta functions
Notes:: See Bellman (1961, pp. 20–24).
Referenced by:: Erratum (V1.1.0) for Subsection 20.10(ii)
Permalink:: http://dlmf.nist.gov/20.10.ii
Correction (effective with 1.1.0):: In the first sentence of this subsection, the constraint $\sinh\left|\beta\right|\leq\ell$ has been replaced with $\left|\Re\beta\right|+\left|\Im\beta\right|\leq\ell$ .
See also:: Annotations for §20.10 and Ch.20

Let $s$ , $\ell$ , and $\beta$ be constants such that $\Re s>0$ , $\ell>0$ , and $\left|\Re\beta\right|+\left|\Im\beta\right|\leq\ell$ . Then

20.10.4		$\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{\beta\pi}{2\ell}\middle\|\frac{i% \pi t}{\ell^{2}}\right)\,\mathrm{d}t=-\int_{0}^{\infty}e^{-st}\theta_{2}\left(% \frac{(\ell+\beta)\pi}{2\ell}\middle\|\frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d% }t=\frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}\right)\operatorname{sech}% \left(\ell\sqrt{s}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $s$ : parameter, $\beta$ : parameter and $\ell$ : parameter Referenced by: Erratum (V1.2.8) for Equation (20.10.4) Permalink: http://dlmf.nist.gov/20.10.E4 Encodings: TeX, pMML, png Correction (effective with 1.2.8): A minus-sign was inserted in front of the second integral and the minus-sign in front of the final term was removed. This is also wrong in Erdélyi et al. (1954a, 4.25(2)). In the second integral, the factor $1+\beta$ was replaced with $\ell+\beta$ . Suggested 2026-08-26 by Zhengxian (Anna) Mei See also: Annotations for §20.10(ii), §20.10 and Ch.20

20.10.5		$\int_{0}^{\infty}e^{-st}\theta_{3}\left(\frac{(\ell+\beta)\pi}{2\ell}\middle\|% \frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{4}% \left(\frac{\beta\pi}{2\ell}\middle\|\frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}% t=\frac{\ell}{\sqrt{s}}\cosh\left(\beta\sqrt{s}\right)\operatorname{csch}\left% (\ell\sqrt{s}\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $s$ : parameter, $\beta$ : parameter and $\ell$ : parameter Referenced by: Erratum (V1.2.8) for Equation (20.10.5) Permalink: http://dlmf.nist.gov/20.10.E5 Encodings: TeX, pMML, png Correction (effective with 1.2.8): In the first integral, the factor $1+\beta$ was replaced with $\ell+\beta$ . Suggested 2026-08-26 by Zhengxian (Anna) Mei See also: Annotations for §20.10(ii), §20.10 and Ch.20

For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193).

§20.10(iii) Compendia

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Permalink:: http://dlmf.nist.gov/20.10.iii
See also:: Annotations for §20.10 and Ch.20

For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2015, §6.16).

20.9 Relations to Other Functions 20.11 Generalizations and Analogs

NIST

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§20.10 Integrals

Contents

§20.10(i) Mellin Transforms with respect to the Lattice Parameter

§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

§20.10(iii) Compendia